CONTAINING TWO·PORTS AND COUPLED TWO.POLES

CONTAINING TWO·PORTS AND COUPLED TWO.POLES

By G. FODOR

Department of Theoretical Electricity. Technical uniyersity. Budapest (Received June 14, 1973)

In a former publication [1] it has heen demonstrated that by character- lzmg two-ports of the network by the chain matrices, a homogeneous descrip- tion can be given, valid also for degenerated t,\-o-ports. While there the cal- culation of the stationary state was considered, now the method will he em- ployed for examinations in the time domain.

1. Stating the problem

Consider a linear time invariant network. Network parameters, the time function of source quantities and the initial (t

=

+0) or starting (t

=

0) values are given. The time function of branch volt ages and branch currents is required.

Our basic aim is to construct the state equation:

~(t) Ax(t) Bz(t) , (1)

where x is the column matrix of the state variables, z that of the source quan- tities, while

A

and

E

are matrices characterizing the network. Including the other variables into a matrix y, they can he expressed in terms of the state variables and the exeitations as:

y(t)

=

Cx(t) Dz(t). (2)

The problem is, on the one hand, the selection of the state variables x, on the other hand the elaboration of a systematic process for the determination of matrices

A, E, C, D

and of Eqs (1) and (2).

2. The network elements

The examined net'work may contain the following lineal' invariant ele- ments (the symbols in hrackets denote both the respective element and the

6 Periodic. Polytechnic. El. 17/4.

334 G. FODOR

number of the respective element; the meaning of the capitals will be clarified later).

a) Voltage sources (v

=

T/l b) Current sources (a

=

A) c) Resistances (r

=

R

+

^G)

d) Condensers (c = C

+

D)

e) Uncoupled or coupled inductors (I

=

L

+

T) f) Resistive two-ports (p P

+ Q,

s

=

S

+

T, p

=

s).

The voltage sources are assumed not to form a loop, the current sources not to form a cut-set, further the two-ports are regarded two-ports with justification (they are not general four-poles).

The individual elements are characterized by the following equations:

where

f

and g are given functions;

'where G"

=

1jRI;;

ick

=

Cl;

u

cI:

lln = L" ill: and llll:

(3)

(4) (5) (6)

(7)

where p indicates the primary port of the two-port, while s the secondary port. (For details see [1].) Parameters denoted by a capital letter are time- independent constants.

3. The state variahles

As known, state variahles of linear invariant networks are very simple to choose, if there are no capacitive (c

+

^v) loops and inductive (I a) cut-sets.

In this case the state variables are the voltage of condensers and the current of inductors.

In a more general case there are two ppssihilities. On the one hand, independent condenser voltages and independent inductor currents can he regarded as state variables. This means that in each independent capacitive loop the voltage of one of the condensers, and in each independent inductive cut-set the current of one of the inductors is not regarded as state variahle.

Let C and Ddenote those condensers, the voltage of which is, or is not a state variahle, respectively, and similarly Land T those coils, the current of 'which

is, or is not a state variable, respectively (c

=

C

+

^{D, 1= L r). In this}

case a proper tree can he chosen in the graph of the network in such a way that all C and

r

branches are twigs, and all D and L hranches are links.

On the other hand, the charge of the cut-iSeLiS generated hy twigs C and the flux at loops generated hy links L, or the volt ages and currents, respectively, proportional to these, can also he chosen as state variahles.

The first mentioned method is simpler, though it has the disadvantage that the continuity of state variahles is not ensul'f~d if the time-function of the excitations contains jumps.

For the sake of comprehensihility, the case without capacitive loops or inductive cut-sets will he examined first, thereafter the more geneTal case.

4. The "regular" network The Kirchhoff laws for voltages and currents are:

Bu

=

0, Qi 0 (8)

where B denotes the loop matrix and Q the cut-set matrix. If the fundamental loop and cut-set system generated hy a tree is chosen, then, as known,

B = [F 1], Q

=

[1 E], E

=

(9)

wheTe 1 is the unit matrix and ^T denotes the transpose.

Let us first examine the case wheTe there are neither capacitive loops, nOT inductive cut-sets in the network. Choose a tree in the following way.

All v and c }Jl'anches are twigs (v

=

V, c

=

C, D

=

0), all a and I hranches aTe links (a A, I = L,

r

⁼ 0), what is possihle, according to the condition.

We should select maximal numher of r hranches as twigs (R-hranches) and minimal numher as links (G-branches). Among the p-hranches (pTimary ports), in turn, possibly minimalnumheT should he chosen as twigs (P-hranches) and maximal numheT as links (Q-hranches). Among the s-bTanches (secondary pOTtS) those designed as t'wigs are denoted hy S, those designed as links hy T.

AccOTdingly, the fOTm of matrix F interpreted in (9), partitioned to hlocks is found to he

.- F.-'\\, F LV

F FQv

F_TV LF_GV V The structuTc of matrix E etc.).

F_{AC FAR PAS} F^{AP -,}

AI

F_{LC FLR F LS F LP} L

Fqc FQR FQS Fop Q

(OP'

Ji\c F-fR F TS FTP T

FGC FGR FGS F GP -' G 0 R S P -(-cut-sets -F+ already follo'ws fTom this (EVA

(10)

1^'~

- 'Av,

336 ^G. FODOR

It follows from the structure of the trce that there are no S- and P-twigs in the loops generated hy the G-hranches (resistive links), and no T-links in the cut-sets generated by the P-branches (primary t'wigs), accordingly

FGS 0, F GP 0, F TP 0,

EsG=O, EpG=O, EpT=O. (ll)

The first group of Kirchhoff laws, on the basis of A-loops and V-cut-sets is found to he

uA. = - (l~.w Uv T FA.c ^Uc FAR uR FAS Us ^11~4P up) , iv -(E~"A~4 -7- EVLiL-T-EvQiQ+EvTiT+EVGiG)'

(12)

These can he calculated in knowledge of the quantities in the right-hand side.

The second group of Kirchhoff laws is given hy the loops

Q,

T, and G, and cut- sets R, S, and P. Taking also (ll) into consideration,

UTTFTRUR FTSus

uG FGR uR

(FQv Uv

+

FQcue) ,

(FTV Uv FTC ue),

(FGV Uv

+

FGcuc) ,

iR T ERQiQ T ERTiT is ESQiQ --;- EST iT

ip

+

EpQiQ

ERG iG

= --

(ERAiA ERLid, (ESA iA

-+-

ESL

id ,

Branch rules (4) and (7) can he written in the following form:

UR = RiR , ^iG Gu_{G ,}

[{: j ^lUPs

11PT Hps

I{PT

1

U; 1

HQS HQT HQS HQT llIps T~IPT ·!'IT ps _~VPT

.]lQS lllQT lVQS l\-QT

(13)

(Ha)

(14h)

Matrices Rand G arc diagonal, ^'W hile hlocks H, H, 1l1, and N contain only one non-zero clement in each row and column.

Upon suhstituting Eqs (14) into (13) 'we obtain (r

+

^p

+

^s) equations for the same numher of unkno'wn quatities i_R, u_G' us' UT' is, iT' thus these can be expressed by means of the given excitations (uv' ⁱA ) and state variables (uc, i_{L ).} These linear functions, further the linear functions "which can be pro- duced by substituting the former into (12), represent the linear function given under (2).

The equations on loops L and cut-sets Care

(15)

The branch rules (5) and (6) are

U L

=

LIL' (16)

where C is diagonal, but L is diagonal only in the ease where there are no coupled inductors. If L is not singular then on the basis of (16) and (15)

FLRUR

+

^FLSus ₍₁₇₎

UC= ^{( ' - I} (E . . E .

, CA lA -t- CL 1L

Substituting the linear functions obtained by solving (13) and (14), state equation (1) has been produced.

5. The generalized network (1st method)

Let us now consider the more general case, namely 'where the net work can contain both capacitive loops and inductiye cut-sets.

In choosing the tree, all v-branehes are chosen as twigs, all a-branches as links (v = V, a = A). Hereafter a maximum luimher of c-branch!'s are ehosen as twigs (C-branches), while the rest become links (D-hranehes).

Similarly a maximum number of I-branches are ehosen as links (L-brunches), while the rest become twigs (r-branches). The same is done for r, p and S

branches. For the sake of understanding, the symbols of branches are tabulated.

Table 1

Branch Twig Link

v V 11\1 =f(t)

a A. i.,'I = g(t)

c C lie ^Cee ^ilC

D liD C_{DD liD}

L

r

^{filL = LLL iL LLr}

^ir

lllr=LrL i~ ^T Lrr i'r

r R G (14a)

p p

s S

Q }(14b)

T

A systematic method of selecting the maximum number of twigs is the following. Consider the partial graphs v, v c, V --L c r o n e by one. (The other branches are substituted by open circuits.) Branches eonsidered pre- viously as twigs are completed to a forest, thus we obtain branches C and R, while the remaining branches become branches D and G, respectively. The

338 G. FODOR

maXImum number of links can he formed analogously by exammmg partial graphs a, a 1, a -;- I p (hut in this case the other hranches are to be sub- stituted by a short-circuit).

The structure of matrix F (upon immediately considering the zero blocks) is now the following:

r FA\" F_{AC F"'R} F AS F AP FAT ^-I F LI· F_{LC FLR} F_{LS F LP} F_Lr P_{Q\. Fqe FQR} F F_Qp 0

(13)

}<' = QS

F_TV F_{TC FTR} fiTS 0 0

FO\! Foc FOR 0 0 0

F DV F Dc 0 ⁽⁾ 0 0 _-1

Choose nO\\· \-oltages ^Uc and currents ^iL as state variables. The construc- tion of the state equation can be performed in the same way as previously.

On the hasis of loops A and cut-spts V, u_A and iF can he expressed in terms of the other variables. On the basis ofloops Q, T, and G, further of cut-sets R, S, and P, as ,\·ell as of Eqs (14) the yoltages and currents with the indicated sub- scripts can he pxpressed by the state yariables and exeitations. On the basis of loops D and cut-sets T, u_D and ir can be expressed by the state variables and excitations, and so can iD and UT hy means of relationships giyen in the table. Finally, on thp hasis of loops L and cut-sets C, and eliminating the previously expressed variahles ,rc obtain the state equation.

6. The generalized network (2nd method)

Let ns choose the tree and the indiyidual branches in the manner de- scribed prcYiollsly. Let the state variables be

U _~ ^Ue -,- Cc~ ECD ('DD U D '

(19)

(We have assumed that LLL is not singular.) It can be easily conceived that Cccu is the total charge of cut-sets generated by branches C, and LLLi is the total flux of loops gcnerated hy branches L. Cut-set charge and loop flux are continuous in thc case of bounded excitation, thus ^U and i are themselves continuous.

The equations 011 loops D and cut-scts T arc

UD FDF UF

ir

+

ErAiA

o

O. ⁽²⁰⁾

Substituting the expressions for u c and iL from (19), ·we have:

(21 ) where

(22) (If there are no coupled coiles, then En En and LLr

=

Ln

=

0.) Substituting variables u_D and ir from (20) into (19), we obtain:

( - )-1 (- )

lIc

=

^{.1 - ECD} F DC . ECD F DV lIv --'-1I ,

(23) where

ECD (24)

The state equation can he produced in thc '.my descrihed under item 5, since lIc and iL are expressed by the state yariahles and excitations according to (23).

In writing the state equation, the terms u_L FLr lIr and ic ECD iD occur in the equations pertaining to loops L and cut-sets C, respectively. These can he expressed simply in terms of the state variables, according to (19) in the forms LLL i and Ccc

n,

respectively. Accordingly, the derivatives of excita- tions are not figuring here and just this ensures that net) and i(t) are continuous in the case of bounded excitation. 'Vith thr 1st method this is not ensuTrd.

7. On the solvahility of the equations

For the sake of completeness it should be mentioned that our process for choosing the state variahles and for constructing the state equations takes only the topology of the network into consideration. In the ease of degenerate two-ports, however, it may happen that the independellce of the designated state variables and the soh-ability of equations (the inn~rtibility of matriees) is not ensured.

A problem may arise if hoth a primary and a seeondary quantity of some of the two-ports are state variables, and the two-port itself is degenerate.

It is easy to prove that if some of the two-port parameters are zero, then some termination pairs are not permitted, in this case one of the state variables

Fig. 1

-

's

Us~

340 G. FODOR

would determine the other, hence only one of them can be regarded as state variable. The forbidden terminations are the following (Fig. 1):

Parameter Primary Secondary Relationship

H 0 Cl L~ Ut = ^-Ki~

K = ⁰ Cl C~ Ul ⁼ HU2

JI = 0 Lt L~ il ^]Vi~

]V 0 Ll C~ il -lvlu₂

In the following the permitted termination pairs are summarized for the typical degenerate two-ports:

:\"ullor (H = O. K = 0, M = O. N = 0) Voltage controlled voltage source (H "'" 0) Voltage controlled current source (K"", 0) Current controlled voltage source (JJ =' 0) Current controlled current source (,'V "'" 0) Ideal transformer

} (H "'" 0, N =' 0) :\" ega tive cOllverter

Gyrator (K =' 0, }[ "'" 0)

Primary

C C

L L

J

^L

l c r

L

l c

Secondary

L

C

L

C C L

L

c

The construction of the state equation by the described method is ensured only if the gIven conditions are satisfied.

Summary

A method is given for the systematic construction of the state equation of linear net- works containing sources, condensers, uncoupled and coupled inductors, resistors, and resistive two-ports. The two-ports can be degenerate too. The cases where there are, or are not capacitive loops and inductive cut-sets in the network, are examined separately.

Reference

1. FODoR. G.: The Analysis of Linear Networks Containing Two-ports and Coupled Two-poles·

Pcriodica Polytechnica. Electr. Eng .• 17, 321- 332 (1973).

Further references see there.

Prof. Dr. Gyorgy FODOR, 1502 Budapest, P. O. B. 91, Hungarl